Questions tagged [ag.algebraic-geometry]

for questions on algebraic geometry, including algebraic varieties, stacks, sheaves, schemes, moduli spaces, complex geometry, quantum cohomology.

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1answer
257 views

intersection cohomology when the resolution is not semi-small

When we have a variety and a resolution of singularities, but it is not semi-small (i.e. the dimensions of the fibres do not satisfy the right conditions), then what can we say about the intersection ...
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7answers
3k views

Is an algebraic space group always a scheme?

Suppose G is a group object in the category of algebraic spaces (over a field, if you like, or even over ℂ if you really want). Is G necessarily a scheme? My feeling is that the answer is "yes" ...
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2answers
654 views

Building elliptic curves into a family

Suppose $E/ \mathbb{Q}$ is an elliptic curve whose Mordell-Weil group $E(\mathbb{Q})$ has rank r. When can we realize E as a fiber of an elliptic surface $S\to C$ fibered over some curve, with ...
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1answer
660 views

Corank 4 hypersurface singularities

A function f: ($\mathbb{C}^n$,0) $\to$ ($\mathbb{C}$,0) is considered a hypersurface singularity if the point $(0,0,\dots,0)$ is the only point in the ideal $\langle \frac{\partial f}{\partial x_1}, \...
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1answer
406 views

Is there a way to check if a relative Hilbert Scheme is reduced?

More specifically, suppose I have a rational curve on a complete intersection, and I know that the relative Hilbert Scheme is not smooth at the point corresponding to my rational curve. Is there any ...
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2answers
2k views

cohomology groups of tensor product of sheaves

If L and M are two local systems on a space X, what can we say about the cohomology groups $H^i(X,L\otimes M)$ in terms of the cohomology of L and M? For example, can we determine their dimensions. ...
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4answers
4k views

Why do automorphism groups of algebraic varieties have natural algebraic group structure?

I am not sure that all automorphism groups of algebraic varieties have natrual algebraic group structure. But if the automorphism group of a variety has algebraic group structure, how do I know the ...
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2answers
209 views

Sheaf isomorphism.

Suppose you have a curve $C$ such that deg$K_C =0$ and $\Gamma(C,\Omega_C^1) \neq 0$. Does this automatically imply that $\vartheta_C \equiv \Omega_C^1$? My thought is yes, I've seen a proposition (...
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2answers
822 views

Is every subgroup of an algebraic group a stabilizer for some action?

Suppose G is an algebraic group (over a field, say; maybe even over ℂ) and H⊆G is a closed subgroup. Does there necessarily exist an action of G on a scheme X and a point x∈X such that ...
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2answers
946 views

Fukaya categories of hyperkahler reductions: general request for information

I'd really like to hear any references or information people have about the Fukaya categories of hyperkahler reductions of vector spaces (for more informations on the varieties, see Nick Proudfoot's ...
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1answer
342 views

finding the closure when blowing a variety at a singularity

I'm having trouble finding the closure in the definition of a blow-up. For example, take the following example, with a node at $(0,0)$ (and at some other points) (it's not a homework question, just a ...
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1answer
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Topological version of Bogomolov’s question

I'm quoting a question from p. 753 of Gromov's recent paper Singularities, Expanders and Topology of Maps: "Does there exist, for every closed oriented $n$-manifold $X_0$, a closed oriented $n$-...
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2answers
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Higher genus closed string B-model

The closed string A-model is mathematically described by Gromov-Witten invariants of a compact symplectic manifold $X$. The genus 0 GW invariants give the structure of quantum cohomology of $X$, which ...
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2answers
962 views

Is the tangent space functor from PD formal groups to Lie algebras an equivalence?

The previous version of this question was rather badly broken, and I hope this version makes some sense. There have been a few questions on MathOverflow about how much representation-theoretic ...
7
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4answers
5k views

Easiest way to determine the singular locus of projective variety & resolution of singularities

For an affine variety, I know how to compute the set of singular points by simply looking at the points where the Jacobian matrix for the set of defining equations has too small a rank. But what is ...
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2answers
363 views

Points on algebraic stacks

I'm a bit confused concerning a definition in Laumon--Moret-Bailly. Perhaps someone could shed some light on the following. It concerns the definition of (closed) point in Chapter 5. More precisely, ...
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6answers
3k views

Are all polynomial inequalities deducible from the trivial inequality?

I remember learning some years ago of a theorem to the effect that if a polynomial $p(x_1, ... x_n)$ with real coefficients is non-negative on $\mathbb{R}^n$, then it is a sum of squares of ...
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1answer
397 views

Semiclassical explanation of “Structured” spaces [closed]

We have our general notions of manifolds, schemes, et cetera, and other geometric "spaces", and we realize that a lot of these look like topological spaces with structure sheaves i.e. structured ...
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4answers
2k views

Finding divisors on a curve

What is the best way to find an actual divisor of an affine curve? I.E. if I am interested in finding a canonical divisor of a curve in two variables, is there a general way to go about it? Do I need ...
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2answers
375 views

Reps of $U(n)$ for the bundles of holomorphic and antiholomorphic forms of projective space

What are the representations of $U(n)$ that induce (see link text) the bundles of holomorphic $\Omega ^{(1,0)}$ and antiholomorphic $\Omega ^{(0,1)}$ forms of $\mathbb{CP}^n$ (recalling the well-known ...
7
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3answers
488 views

Weierstrass points on rigid-analytic surfaces

Does a rigid-analytic surface defined over a nonarchimedean complete field have Weierstrass points (if its genus is big enough let's say)? Is there a good reference that (ideally) lists theorems for ...
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4answers
4k views

Definition of étale for rings

Let $A \to B$ be a ring extension. What is the definition of $B/A$ étale ? When $A$ is a field, do we get a nice characterization ?
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2answers
6k views

Projective closure of affine curve

Is there a generalized method to find the projective closure of an affine curve? For example, I read that the projective closure of $y^2 = x^3−x+1$ in $\mathbb{P}^2$ is $y^2z = x^3−xz^2+z^3$. If I ...
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4answers
1k views

Near Trivial Quiver Varieties

So, today I started learning the definition of a quiver variety, and wanted to make sure I'm understanding things right, so first, my setup: I've been looking at the simplest case that didn't look ...
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2answers
310 views

A technical question about derivations of sheaves on group schemes

Let $G$ be a group scheme (for instance, over $k$ a field of characteristic 0). Let $e$ be its unit. I denote by $O_G$ the structural sheaf of $G$. Let $D_e : O_{G,e} \to k$ a derivation. I would ...
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3answers
704 views

Line bundles trivial after extension of the base-field

Let k be a field and let X be scheme over k. Let K be a field extension of k and denote by $X_K$ the base-change of X to Spec K. Under what conditions is the canonical map of Picard groups $Pic(X)\to ...
35
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4answers
5k views

What does “linearly disjoint” mean for abstract field extensions?

All definitions I've seen for the statement "$E,F$ are linearly disjoint extensions of $k$" are only meaningful when $E,F$ are given as subfields of a larger field, say $K$. I am happy with the ...
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2answers
3k views

Why is the decomposition theorem awesome?

I saw the statement of the decomposition theorem for perverse sheaves sometime ago. I know that (modulo most of the details) it implies some big theorems in algebraic geometry and gives new proofs for ...
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3answers
1k views

Holomorphic and antiholomorphic forms of projective space

For $\mathbb{CP}^1$ the bundles of holomorphic and antiholomorphic forms are equal to the $\mathcal{O}(-2)$ and $\mathcal{O}(2)$ respectively. Do the holomorphic and antiholomorphic bundles of $\...
3
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3answers
438 views

Nature of Invertible Sheaves in which there are no global sections.

EDIT: Let me try to make the question clearer. Consider the invertible sheaves $\mathcal{O}(d)$ over the projective space $\mathbb{P}^n$ where $d\in \mathbb{Z}$. Now, if $d>0$, among many ...
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10answers
3k views

How can I really motivate the Zariski topology on a scheme?

First of all, I am aware of the questions about the Zariski topology asked here and I am also aware of the discussion at the Secret Blogging Seminar. But I could not find an answer to a question that ...
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1answer
309 views

Systems of conics

It seems well-known that the system of conics given by $\frac{x^2}{a^2}+\frac{y^2}{a^2-c^2}=1$ for $c>0$ fixed and $a \in (0,c)\cup(c,\infty)$ varying is orthogonal: whenever two of these curves ...
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5answers
2k views

Elliptic Curves over F_1?

Is there an notion of elliptic curve over the field with one element? As I learned from a previous question, there are several different versions of what the field with one element and what schemes ...
3
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4answers
990 views

Examples of divisors on an analytical manifold

I am trying to understand divisors reading through Griffith and Harris but it is difficult to come up with any particular interesting example. I have browsed through Hartshone's book but everything is ...
4
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2answers
747 views

a question about Gromov-Witten invariant

Do the Gromov-Witten invariants count the morphisms from a curve to a variety over $\mathbb{C}$?
4
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2answers
212 views

On the dimension of moduli space of pointed curves with fixed Weierstrass semigroup

Does anyone know any information on the question of the dimension of moduli space of pointed curves with fixed Weierstrass semigroup? Some conjecture?
20
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4answers
3k views

What is a section?

This question comes out of the answers to Ho Chung Siu's question about vector bundles. Based on my reading, it seems that the definition of the term "section" went through several phases of ...
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3answers
935 views

Why a subvariety of a variety of general type is of general type

It seems a well-known fact that subvarieties of a variety of general type containing a general point are also of general type. This fact is an essential property used to prove some extension theorems ...
19
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3answers
2k views

Elkies' supersingularity theorem in higher dimension

The following is a theorem of Elkies: Let $X$ be an elliptic curve over $\mathbb{Q}$. Then there are infinitely many primes $p$ such that the action of Frobenius on $H^1(\mathcal{O}, X)$ is zero. ...
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4answers
385 views

Is tensoring with a module representable iff it is locally free of finite rank?

Motivation: It's nice when you can think of the elements of an $A$-module $M$ as sections some $A$-scheme $Y\to Spec(A)$. That is, maps $Spec(A)\to Y$ such that $Spec(A)\to Y \to Spec(A)$ is the ...
0
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3answers
909 views

About the intersection of two vector bundles

I´m looking for information about the intersection of two vector bundles (principally trivial bundles, but no necessarily). I´m trying to make a picture (literally) of reflexive finite generated ...
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11answers
9k views

Non-commutative algebraic geometry

Suppose I tried to take Hartshorne chapter II and re-do all of it with non-commutative rings rather than commutative rings. Is this possible? Which parts work in the non-commutative setting and which ...
4
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1answer
704 views

Torsion line bundles with non-vanishing cohomology on smooth ACM surfaces

I am looking for an example of a smooth surface $X$ with a fixed very ample $\mathcal O_X(1)$ such that $H^1(\mathcal O(k))=0$ for all $k$ (such thing is called an ACM surface, I think) and a globally ...
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3answers
1k views

Why is a variety of general type hyperbolic?

I heard people mentioned this in one sentence, but don't see the reason. Why a (smooth) variety of general type, i.e. an algebraic variety X with K_X big, is hyperbolic, i.e. has no non-constant map ...
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8answers
9k views

Why is it useful to study vector bundles?

I have this question coming from an earlier Qiaochu's post. Some answers there, especially David Lehavi's one, were drawing the analogy bundles and varieties versus modules and rings. So I am just ...
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4answers
1k views

Negative Gromov-Witten invariants

I understand the heuristic reason why Gromov-Witten invariants can be rational; roughly it's because we're doing curve counts in some stacky sense, so each curve $C$ contributes $1/|\text{Aut}(C)|$ to ...
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5answers
2k views

Intuition behind moduli space of curves

For a genus g compact smooth surface $M$, an algebraic structure is the same as a complex structure is the same as a conformal structure. So the moduli space of smooth curves should be the same as the ...
21
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1answer
786 views

Do DG-algebras have any sensible notion of integral closure?

Suppose R → S is a map of commutative differential graded algebras over a field of characteristic zero. Under what conditions can we say that there is a factorization R → R' → S ...
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3answers
3k views

What is “restriction of scalars” for a torus?

I am starting on a Phd program and am supposed to read Colliot Thelene and Sansuc's article on R-equivalence for tori. I find it very difficult and although I have some knowledge over schemes , I am ...
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7answers
3k views

Why is Riemann-Roch an Index Problem?

I was in a lecture not long ago given by C. Teleman and at some point he said "Well, since Riemann-Roch is an index problem we can do..." Then right after that he argued in favour of such a sentence. ...

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